Method for Modeling Interfaces

Abstract

Simulations of interfacial phenomena are useful to understand and predict the phase-change processes that occur around the interface at various operating conditions. A method is described that simplifies the simulation of interfaces and the transport processes occurring across the interfaces. The method accurately determines the fixed ϕ value at the computational cells that have an interface (mixture cells), which is needed for a proper representation of the interface. Moreover, the method accurately determines the gradient of ϕ at the interface, which is needed for a proper estimation of the mass transfer at the interface.

What is claimed: 
     
         1 . A method for modeling interfaces comprises:
 determining a fixed value of a variable ϕ at mixture cells for a representation of an interface; and   determining the mass transfer through the interface with the gradient of ϕ at the interface, wherein the gradient is computed on a phase that is driving the mass transfer, wherein fixing the value of the mixture cells, the gradient is computed on the phase driving the mass transfer, by evaluating the gradient of the variable ϕ by performing an algorithm that identifies the center of cell in the phase that is closest to the interface in the normal direction, the algorithm injecting three proves in a direction perpendicular to the interface, and the fixed value of the variable ϕ and the interfacial mass transport are found with the information of the length of the probes and the values of the variable ϕ at the extremes of the probes, wherein mathematical expressions are derived based on the gradients of the variable ϕ at two different locations at the interface, such that a representation of the conditions at the interface in a multiphase simulation and computation of the mass transfer at the interface are achieved.   
     
     
         2 . A one-cell method for evaluating the temperature gradient at the interface, comprising:
 1) identifying a mixture cell for which a fixed ϕ value or the mass transfer is to be determined;   2) injecting a probe-1, wherein the origin is the mixture cell center, the orientation is normal to the interface, and the length is such that the tip intersects the interface such that the point intersects the probe-1 with the interface at a point a;   3) injecting a probe-2, wherein the origin is the point a, the orientation is normal to the interface, and the length is d 2 ;   4) exploring a region around the tip of the probe-2 to find the center of an I-cell that is nearest to the interface in the normal direction;   5) injecting a probe-3, wherein the origin is the cell center of the I-cell, the orientation is normal to the interface, and the length is determined by the following steps 6) and 7);   6) creating a vector {right arrow over (r I )} connecting the center of the I-cell with the point a at the interface; and   7) determining the length of the probe-3 by the projection of the vector {right arrow over (r I )} on the normal vector n, and determining the intersection point b between the probe-3 and the interface.   
     
     
         3 . A method comprising:
 injecting three probes that are perpendicular to the interface, wherein probe-1 supports the injection of probe-2, probe-1 and probe-2 support the construction of probe-3 and probe-3 is normal to the interface, and has a length such that the tip of the probe intersects a cell center on a phase;   injecting probe-1 based on the identification of the computational cell that has an interface, wherein the origin of probe-1 is the mixture cell center, the orientation is normal to the interface, and the tip of the probe is the intersection of the probe with the interface, wherein the point that intersects the probe with the interface can be identified as point a, the length of the probe-1 is given by d 1 , and the length of the probe is determined based on the origin and the tip of probe-1;   injecting probe-2 based on the tip of the probe-1 point a, wherein the origin of probe-2 is point a, the orientation is normal to the interface, and the length is d 2 ;   using probe-2 to identify the center of computational cell in a phase that is closest to the interface along the normal direction, wherein the nearest cell can be identified as I-cell;   exploring a region that uses the tip of probe-2 as a reference point to find the center of the I-cell that is nearest to the interface along the normal direction, wherein the center of the explored region corresponds to the location of the tip of probe-2, wherein the dimensions of the explored region are Δx in the x-direction, Δy in the y-direction, and Δz in the z-direction, where x, y, and z represent the coordinates of a Cartesian system and Δx, Δy, and Δz are the lengths of the computational cells in the x, y, and z, directions, respectively;   using the I-cell to inject the probe-3, wherein the origin of the probe-3 is the center of the I-cell, the orientation is normal to the interface, and the length is determined by the projection or dot-product of the vector {right arrow over (r I )} on the unit vector that is normal to the interface {circumflex over (n)}, wherein (d 3 ={right arrow over (r I )} ·{circumflex over (n)}). {right arrow over (r I )} is a vector that connects the center of the I-cell with the point a at the interface;   using the lengths of probe-1 “d 1 ” and probe “d 3 ” and the values of the variable ϕ at the extremes of the probes to determine the value ϕ M  of the computational cells that have an interface;   determining the fixed value ϕ M  by adding the product of the length of the probe-1 “d 1 ” with the gradient of the variable ϕ at point a to the value of the variable ϕ at the interface, wherein ϕ M =ϕ in +d 1 (ϕ I −ϕ in )/d 3 . ϕ in  is the value of the variable ϕ at the interface and ϕ I  is the value of the variable ϕ inside the computational cell-I;   using the lengths of probe-1 “d 1 ” and probe “d 3 ” and the values of the variable ϕ at the extremes of the probes to determine the transport of mass or species or any other transport across the interface;   determining the transport of mass or species m″ by the product of a parameter α and the gradient of the variable ϕ at point “a”, m″=α(ϕ I −ϕ in )/d 3 , where a is constant or a variable parameter that depends on the properties of the fluids or phases in the simulation;   estimating the length of the probe-2 based on the dimensions of the mixture-cell, wherein the length of probe-2 d 2  is such that the tip of probe-2 lies outside of the mixture cell; and   identifying the center of the I-cell based on the tip of probe-2 and an explored region, wherein the length of the explored region is given the dimensions of the computational cells (Δx, Δy, and Δz) or by any other parameters.